\documentclass{article}%
\usepackage{graphicx}
\usepackage{amsmath}%
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\begin{document}
La expresi\'{o}n $\dfrac{y^{3}}{x^{2}-a^{4}y^{2}}-\dfrac{a}{x+a^{2}y}%
+\dfrac{xy}{x-a^{2}y}$ se puede simplificar como:\newline\qquad a)
$\dfrac{x^{2}y-xa+xy^{2}a^{2}+a^{3}y+y^{3}}{\left(  x-a^{2}y\right)  \left(
x+a^{2}y\right)  }\qquad$b) $\dfrac{x^{2}y-xa+xy^{2}a^{2}+a^{3}y-y^{3}%
}{\left(  x-a^{2}y\right)  \left(  x+a^{2}y\right)  }$\newline\qquad c)
$\dfrac{x^{2}y-xa-xy^{2}a^{2}+a^{3}y+y^{3}}{\left(  x-a^{2}y\right)  \left(
x+a^{2}y\right)  }\qquad$d) $\dfrac{x^{2}y-xa+xy^{2}a^{2}-a^{3}y+y^{3}%
}{\left(  x-a^{2}y\right)  \left(  x+a^{2}y\right)  }$

La expresi\'{o}n $\dfrac{2y^{3}}{x^{2}-4a^{4}y^{2}}-\dfrac{4a}{x+2a^{2}%
y}+\dfrac{xy}{x-2a^{2}y}$ se puede simplificar como:\newline\qquad a)
$\dfrac{x^{2}y-4xa+2xy^{2}a^{2}+8a^{3}y+2y^{3}}{\left(  x-2a^{2}y\right)
\left(  x+2a^{2}y\right)  }\qquad$b) $\dfrac{x^{2}y+4xa+2xy^{2}a^{2}%
-8a^{3}y+2y^{3}}{\left(  x-2^{2}y\right)  \left(  x+2a^{2}y\right)  }$%
\newline\qquad c) $\dfrac{x^{2}y-4xa-2xy^{2}a^{2}+8a^{3}y-2y^{3}}{\left(
x-2^{2}y\right)  \left(  x+2a^{2}y\right)  }\qquad$d) $\dfrac{x^{2}%
y+4xa+2xy^{2}a^{2}+8a^{3}y+2y^{3}}{\left(  x-2^{2}y\right)  \left(
x+2a^{2}y\right)  }$

La expresi\'{o}n $\dfrac{y^{3}}{x^{2}-4a^{4}y^{2}}-\dfrac{4a}{x+2a^{2}%
y}+\dfrac{5xy}{x-2a^{2}y}$ se puede simplificar como:\newline\qquad a) $-$
$\dfrac{y^{3}+10xa^{2}y^{2}+8ya^{3}+5yx^{2}-4xa}{\left(  -x+2a^{2}y\right)
\left(  x+2a^{2}y\right)  }\qquad$b) $-$ $\dfrac{3y^{3}+5xa^{2}y^{2}%
+8ya^{3}+5yx^{2}-4xa}{\left(  -x+2a^{2}y\right)  \left(  x+2a^{2}y\right)  }%
$\newline\qquad c) $-$ $\dfrac{y^{3}+10xa^{2}y^{2}+8ya^{3}+5yx^{2}%
-4xa}{\left(  -x+a^{2}y\right)  \left(  x+a^{2}y\right)  }\qquad$d)
$\dfrac{y^{3}+10xa^{2}y^{2}+8ya^{3}+5yx^{2}-4xa}{\left(  -x+2a^{2}y\right)
\left(  x+2a^{2}y\right)  }$

La expresi\'{o}n $\dfrac{y^{3}}{x^{2}-9a^{2}y^{2}}-\dfrac{4a}{x+3ay}%
+\dfrac{5xy}{x-3ay}$ se puede simplificar como:\newline\qquad a) $\dfrac
{y^{3}+15xay^{2}+12a^{2}y+5yx^{2}-4xa}{\left(  x-3ay\right)  \left(
x+3ay\right)  }$ $\qquad$b) $-\dfrac{y^{3}+15xay^{2}+12a^{2}y+5yx^{2}%
-4xa}{\left(  x-3ay\right)  \left(  x+3ay\right)  }$\newline\qquad c)
$\dfrac{y^{3}+15xay^{2}+12a^{2}y+5yx^{2}-4xa}{\left(  x-9ay\right)  \left(
x+9ay\right)  }\qquad$d) $\dfrac{2y^{3}+3xay^{2}+12a^{2}y+5yx^{2}-4xa}{\left(
x-3ay\right)  \left(  x+3ay\right)  }$

La expresi\'{o}n $\dfrac{y^{4}}{x^{2}-16a^{2}y^{4}}-\dfrac{4a}{x+4ay^{2}%
}+\dfrac{5xy}{x-4ay^{2}}$ se puede simplificar como:\newline\qquad a)
$\dfrac{y^{4}+20xay^{3}+16a^{2}y^{2}+5yx^{2}-4xa}{\left(  x-4ay^{2}\right)
\left(  x+4ay^{2}\right)  }$ $\qquad$b) $\dfrac{y^{4}+20xay^{3}+16a^{2}%
y^{2}+5yx^{2}-4xa}{\left(  x-2ay^{2}\right)  \left(  x+2ay^{2}\right)  }$

\qquad c) $\dfrac{y^{4}+20xay^{3}+16a^{2}y^{2}+5yx^{2}-4xa}{\left(
x-4a^{2}y\right)  \left(  x+4a^{2}y\right)  }\qquad$d) $-\dfrac{y^{4}%
+20xay^{3}+16a^{2}y^{2}+5yx^{2}-4xa}{\left(  x-4ay^{2}\right)  \left(
x+4ay^{2}\right)  }$

La expresi\'{o}n $\dfrac{2y^{4}}{x^{2}-25a^{4}y^{4}}+\dfrac{2a}{x+5a^{2}y^{2}%
}-\dfrac{3xy}{x-5a^{2}y^{2}}\allowbreak\allowbreak\allowbreak$ se puede
simplificar como:\newline\qquad\medskip a) $\dfrac{2y^{4}-15xa^{2}%
y^{3}-10a^{3}y^{2}-3yx^{2}+2xa}{\left(  x-5a^{2}y^{2}\right)  \left(
x+5a^{2}y^{2}\right)  }$\qquad b)$\dfrac{2y^{4}-15xa^{2}y^{3}-10a^{3}%
y^{2}-3yx^{2}+2xa}{\left(  x-5ay^{2}\right)  \left(  x+5ay^{2}\right)  }%
$\newline\qquad c) $\dfrac{2y^{4}-15xa^{2}y^{3}+10a^{3}y^{2}-3yx^{2}%
+2xa}{\left(  x-5a^{2}y^{2}\right)  \left(  x+5a^{2}y^{2}\right)  }$ \qquad d)
$-\dfrac{2y^{4}-15xa^{2}y^{3}-10a^{3}y^{2}-3yx^{2}+2xa}{\left(  x-5a^{2}%
y^{2}\right)  \left(  x+5a^{2}y^{2}\right)  }$

La expresi\'{o}n $\dfrac{y^{4}}{x^{2}-4a^{6}y^{4}}-\dfrac{9a}{x-2a^{3}y^{2}%
}+\dfrac{5xy}{x+2a^{3}y^{2}}\allowbreak\allowbreak$ $\allowbreak$ se puede
simplificar como:\newline\qquad\medskip a) $\dfrac{y^{4}-10xa^{3}y^{3}%
-18a^{4}y^{2}+5yx^{2}-9xa}{\left(  x-2a^{3}y^{2}\right)  \left(  x+2a^{3}%
y^{2}\right)  }$\qquad b) $\dfrac{y^{4}-10xa^{3}y^{3}+18a^{4}y^{2}%
+5yx^{2}-9xa}{\left(  x-2a^{3}y^{2}\right)  \left(  x+2a^{3}y^{2}\right)  }%
$\newline\qquad c) $-\dfrac{y^{4}-10xa^{3}y^{3}-18a^{4}y^{2}+5yx^{2}%
-9xa}{\left(  x-2a^{3}y^{2}\right)  \left(  x+2a^{3}y^{2}\right)  }$ \qquad d)
$\dfrac{y^{4}-10xa^{3}y^{3}-18a^{4}y^{2}+5yx^{2}-9xa}{\left(  x-2ay^{2}%
\right)  \left(  x+2ay^{2}\right)  }$

La expresi\'{o}n $\dfrac{3y^{4}}{x^{2}-9a^{6}y^{2}}+\dfrac{5a}{x-3a^{3}%
y}+\dfrac{4xy}{x+3a^{3}y}$ se puede simplificar como:\newline\qquad\medskip a)
$\dfrac{3y^{4}-12xa^{3}y^{2}+15a^{4}y+4yx^{2}+5xa}{\left(  x-3a^{3}y\right)
\left(  x+3a^{3}y\right)  }$\qquad b) $\dfrac{3y^{4}-12xa^{3}y^{2}%
-15a^{4}y+4yx^{2}+5xa}{\left(  x-3a^{2}y\right)  \left(  x+3a^{2}y\right)  }%
$\newline\qquad c) $\dfrac{3y^{4}-12xay^{2}+15a^{4}y+4yx^{2}+5xa}{\left(
x-3a^{3}y\right)  \left(  x+3a^{3}y\right)  }$ \qquad d) $-\dfrac
{3y^{4}-12xa^{3}y^{2}+15a^{4}y+4yx^{2}+5xa}{\left(  x-3a^{3}y\right)  \left(
x+3a^{3}y\right)  }$

La expresi\'{o}n $\dfrac{y^{4}}{x^{2}-16a^{2}y^{2}}-\dfrac{3a}{x-4ay}%
-\dfrac{2xy}{x+4ay}\allowbreak$ se puede simplificar como:\newline%
\qquad\medskip a) $\dfrac{y^{4}+8xay^{2}-12a^{2}y-2yx^{2}-3xa}{\left(
x-4ay\right)  \left(  x+4ay\right)  }$ \ \ \ \ \ \ \ \ \ \ b) $\dfrac
{y^{4}+8xay^{2}-12a^{2}y-2yx^{2}-3xa}{\left(  x-4a^{2}y\right)  \left(
x+4a^{2}y\right)  }$\newline\qquad c) $\dfrac{y^{4}+8xay^{2}-12a^{2}%
y-2yx-3xa}{\left(  x-4ay\right)  \left(  x+4ay\right)  }$ \qquad d)
$-\dfrac{y^{4}+8xay^{2}-12a^{2}y-2yx^{2}-3xa}{\left(  x-4ay\right)  \left(
x+4ay\right)  }$

La expresi\'{o}n $\dfrac{3y^{4}}{x^{2}-36a^{2}y^{4}}+\dfrac{2a}{x-6ay^{2}%
}-\dfrac{3xy}{x+6ay^{2}}\allowbreak$ se puede simplificar como:\newline%
\qquad\medskip a) $\dfrac{3y^{4}+18xay^{3}+12a^{2}y^{2}-3yx^{2}+2xa}{\left(
x-6ay^{2}\right)  \left(  x+6ay^{2}\right)  }$\qquad b) $\dfrac{3y^{4}%
+18xay^{3}+12a^{2}y^{2}-3yx^{2}+2xa}{\left(  x-6ay\right)  \left(
x+6ay\right)  }$\newline\qquad c) $\dfrac{3y^{4}+18xay^{3}-12a^{2}%
y^{2}-3yx+2xa}{\left(  x-6ay^{2}\right)  \left(  x+6ay^{2}\right)  }$ \qquad
d) $-\dfrac{3y^{4}+18xay^{3}+12a^{2}y^{2}-3yx^{2}+2xa}{\left(  x-6ay^{2}%
\right)  \left(  x+6ay^{2}\right)  }$

La expresi\'{o}n $\dfrac{2y^{4}}{x^{2}-16a^{4}y^{2}}+\dfrac{3a}{x-4a^{2}%
y}+\dfrac{4xy}{x+4a^{2}y}\allowbreak$ se puede simplificar como:\newline%
\qquad\medskip a) $\dfrac{2y^{4}-16xa^{2}y^{2}+12a^{3}y+4yx^{2}+3xa}{\left(
x-4a^{2}y\right)  \left(  x+4a^{2}y\right)  }$\qquad b) $-\dfrac
{2y^{4}-16xa^{2}y^{2}+12a^{3}y+4yx^{2}+3xa}{\left(  x-4a^{2}y\right)  \left(
x+4a^{2}y\right)  }$\newline\qquad c) $\dfrac{2y^{4}-16xa^{2}y^{2}%
+12a^{3}y+4yx-3xa}{\left(  x-4a^{2}y\right)  \left(  x+4a^{2}y\right)  }%
$\qquad d) $\dfrac{2y^{4}-16xa^{2}y^{2}+12a^{3}y+4yx^{2}+3xa}{\left(
x-4ay\right)  \left(  x+4ay\right)  }$

La expresi\'{o}n $\dfrac{3y^{4}}{x^{2}-25a^{6}y^{2}}-\dfrac{2a}{x+5a^{3}%
y}-\dfrac{2xy}{x-5a^{3}y}\allowbreak$ se puede simplificar como:\newline%
\qquad\medskip a) $\dfrac{3y^{4}-10xa^{3}y^{2}+10a^{4}y-2yx^{2}-2xa}{\left(
x-5a^{3}y\right)  \left(  x+5a^{3}y\right)  }$\qquad b) $-\dfrac
{3y^{4}-10xa^{3}y^{2}+10a^{4}y-2yx^{2}-2xa}{\left(  x-5a^{3}y\right)  \left(
x+5a^{3}y\right)  }$\newline\qquad c) $\dfrac{3y^{4}-10xa^{3}y^{2}%
+10ay+2yx^{2}-2xa}{\left(  x-5a^{3}y\right)  \left(  x+5a^{3}y\right)  }$
\qquad d) $\allowbreak\dfrac{3y^{4}-10xa^{3}y^{2}+10a^{4}y-2yx^{2}%
-2xa}{\left(  x-5a^{3}y^{2}\right)  \left(  x+5a^{3}y^{2}\right)  }$

La expresi\'{o}n $\dfrac{y^{4}}{x^{2}-9a^{4}y^{2}}-\dfrac{3a}{x+3a^{2}%
y}+\dfrac{4xy}{x-3a^{2}y}\allowbreak$ se puede simplificar como:\newline%
\qquad\medskip a) $\dfrac{y^{4}+12xa^{2}y^{2}+9a^{3}y+4yx^{2}-3xa}{\left(
x-3a^{2}y\right)  \left(  x+3a^{2}y\right)  }$\qquad b) $\dfrac{y^{4}%
+12xa^{2}y^{2}+9a^{3}y+4yx^{2}-3xa}{\left(  x-3ay\right)  \left(
x+3ay\right)  }$\newline\qquad c) $\dfrac{y^{4}+12xa^{2}y^{2}+9a^{2}%
y-4yx^{2}-3xa}{\left(  x-3a^{2}y\right)  \left(  x+3a^{2}y\right)
}\allowbreak$ \qquad d) $-\dfrac{y^{4}+12xa^{2}y^{2}+9a^{3}y+4yx^{2}%
-3xa}{\left(  x-3a^{2}y\right)  \left(  x+3a^{2}y\right)  }$

La expresi\'{o}n $\dfrac{2y^{4}}{x^{2}-4a^{6}y^{4}}-\dfrac{4a}{x-2a^{3}y^{2}%
}+\dfrac{5xy}{x+2a^{3}y^{2}}\allowbreak$ se puede simplificar como:\newline%
\qquad\medskip a) $\dfrac{2y^{4}-10xa^{3}y^{3}-8a^{4}y^{2}+5yx^{2}%
-4xa}{\left(  x-2a^{3}y^{2}\right)  \left(  x+2a^{3}y^{2}\right)  }$\qquad b)
$-\dfrac{2y^{4}-10xa^{3}y^{3}-8a^{4}y^{2}+5yx^{2}-4xa}{\left(  x-2a^{3}%
y^{2}\right)  \left(  x+2a^{3}y^{2}\right)  }$ \newline\qquad c)
$\dfrac{2y^{4}-10xa^{2}y^{3}+8a^{2}y^{2}+5yx^{2}-4xa}{\left(  x-2a^{3}%
y^{2}\right)  \left(  x+2a^{3}y^{2}\right)  }$ \qquad d) $\dfrac
{2y^{4}-10xa^{3}y^{3}-8a^{4}y^{2}+5yx^{2}-4xa}{\left(  x-2a^{2}y^{2}\right)
\left(  x+2a^{2}y^{2}\right)  }$

La expresi\'{o}n $\dfrac{4y^{4}}{x^{2}-25a^{2}y^{4}}+\dfrac{2a}{x+5ay^{2}%
}-\dfrac{3xy}{x-5ay^{2}}\allowbreak$ \qquad\ se puede simplificar
como:\newline\qquad\medskip a) $\dfrac{4y^{4}-15xay^{3}-10a^{2}y^{2}%
-3yx^{2}+2xa}{\left(  x-5ay^{2}\right)  \left(  x+5ay^{2}\right)  }$\qquad b)
$-\dfrac{4y^{4}-15xay^{3}-10a^{2}y^{2}-3yx^{2}+2xa}{\left(  x-5ay^{2}\right)
\left(  x+5ay^{2}\right)  }$\newline\qquad c) $-\dfrac{4y^{4}-15xay^{3}%
-10ay^{2}-3yx+2xa}{\left(  x-5ay^{2}\right)  \left(  x+5ay^{2}\right)  }$
\qquad d) $\dfrac{4y^{4}-15xay^{3}-10a^{2}y^{2}-3yx^{2}+2xa}{\left(
x-5ay\right)  \left(  x+5ay\right)  }$


\end{document}